Bifrequency and bispectrum maps: a new look at multirate systems with stochastic inputs

Abstract

In multirate digital signal processing, we often encounter decimators, interpolators, and complicated interconnections of these with LTI filters. We also encounter cyclo-wide-sense stationary (CWSS) processes and linear periodically time-varying (LPTV) systems. It is often necessary to understand the effects of multirate systems on the statistical properties of their input signals. Some of these issues have been addressed earlier. For example, it has been shown that a necessary and sufficient condition for the output of an-fold interpolation filter to be wide sense stationary (WSS) for all WSS inputs is that the filter have an alias-free () support. However, several questions of this nature remain unanswered. For example, what is the necessary and sufficient condition on a pair (or more generally a bank) of interpolation filters so that their outputs are jointly WSS (JWSS) for all jointly WSS inputs? What is the condition if only the sum of their outputs is required to be WSS? When is the output of an LPTV system (for example a uniform filter-bank) WSS for all WSS inputs? Some of these questions may appear to be simple generalizations of the above-mentioned result for a single interpolation filter. However, the frequency domain approaches that proved this result are quite difficult to generalize to answer these questions. The purpose of this paper is to provide these answers using analysis based on bifrequency maps and bispectra. These tools are two-dimensional (2-D) Fourier transforms that characterize all linear time-varying (LTV) systems and nonstationary random processes, respectively. We show that the questions raised above can be addressed elegantly and in a geometrically insightful way using these tools. We also derive a bifrequency characterization of lossless LTV systems. This may potentially lead to an increased understanding of these systems.

Shift variance, cyclostationarity and expected shift variance in multirate LPSV systems

“…which is consistent with bispectrum properties described in [5,2]. In the Fourier domain, SV 2 (T ) thus is…”

“…Multirate systems with FIR filters are LPSV, and their behavior with respect to random input signals has been analyzed in, e.g., [1,2,3]. Although the shift variance of a multirate LPSV system is generated by non-ideal filtering during decimation, while cyclic nonstationarities are caused by non-ideal anti-imaging filtering during interpolation [4], the two phenomena are closely related, and were jointly treated in, e.g., [5,4].…”

The fundamental input/output structure of a linear, time-varying array receiver

“…The aim is to connect intuitively with the impulse-response and frequencyresponse characterizations of LTI systems while both raising the mathematical level enough to make it interesting to (say) an engineering professor and omitting details to respect limitations on her time. (The corresponding discrete-time theory is developed, somewhat more thoroughly, by Akkarakaran and Vaidyanathan [2]. )…”

Minimum variance in fast, slow and dual‐rate control loops

“…It is known that stationary random processes, when passed through multirate systems, become cyclo-stationary [21]. Thus the fast-sampled output of the DR system, y f ; is a cyclo-stationary signal.…”